Problem: The arithmetic sequence $a_i$ is defined by the formula: $a_1 = -4910$ $a_i = a_{i - 1} + 8$ Find the sum of the first $575$ terms in the sequence.
Answer: Getting started Let's write out the first few terms of the series: $-4910 + (-4902) + (-4894) + (-4886)...$ We're dealing with an arithmetic series because the difference between terms is constant. That is, each term is $8$ greater than the one before it. We need a formula to compute the sum of the terms. Formula for arithmetic series The sum $S_n$ of a finite arithmetic series is $S_n = \dfrac {\left(a_1 + a_n \right)}{2} \cdot n$ where $a_1$ is the first term, $a_n$ is the last term, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = {-4910})$ and the number of terms $(n = {575})$ are given in the question. We need to find the last term $(a_n)$. Step 1: Find $a_n$ (the last term) There are $575 -1= 574$ terms after the first term. The sequence increases by $8$ for each new term. So, the sequence increases by a total of $574 \cdot 8 = 4592$ from where it starts at $-4910$. That means the last term must be $-4910+4592 = {-318}$. In other words, $a_n = {-318}$. Step 2: Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac {\left(a_1 + a_n \right)}{2} \cdot n \\\\ S_{{575}}&= \dfrac {\left({-4910} + ({-318}) \right)}{2} \cdot {575} \\\\ S_{{575}} &= -2614 \left(575\right) \\\\ S_{{575}} &= -1{,}503{,}050\end{aligned}$ The answer $ -1{,}503{,}050 $